On the Limitations of a Generalized Vaidya Metric

TL;DR

This paper demonstrates that the generalized Vaidya metric cannot be smoothly matched to an exterior Schwarzschild or Vaidya solution unless the mass function depends solely on the null coordinate, highlighting limitations in modeling bounded stellar objects.

Contribution

It provides a rigorous proof that the generalized Vaidya spacetime cannot serve as a bounded stellar interior with a regular surface unless the mass function is only a function of the null coordinate.

Findings

Discontinuity in extrinsic curvature induces a surface stress-energy tensor.

Mismatch in quasi-local energy flux across boundary.

Generalized Vaidya spacetime is better interpreted as unbounded with heat flux.

Abstract

We prove that there can not be a smooth matching of the Generalized Vaidya metric with an exterior Schwarzschild/Vaidya patch across a finite boundary hypersurface unless the mass function is a function of the null coordinate alone. By explicitly deriving the extrinsic curvature components, we show that for $\partial m / \partial r \neq 0$ one has a discontinuity in the curvature and induces a surface stress-energy tensor, corresponding to a thin shell of matter. This discontinuity also appears in the geometric invariant $\mathcal{K} = K_{ab}K^{ab}$ and in the Kodama current, indicating a mismatch in quasi-local energy flux across the boundary. The analysis of timelike geodesics leads to the same condition, reinforcing that the generalized Vaidya geometry with $\partial m / \partial r \neq 0$ cannot represent a consistent stellar interior bounded by a regular surface. We therefore note that the generalized Vaidya spacetime should be interpreted as an unbounded geometry with intrinsic heat flux rather than a viable bounded source.